Uniqueness and nonuniqueness of p-harmonic Green functions on weighted R and metric spaces

Abstract

We study uniqueness of p-harmonic Green functions in domains Ω in a complete metric space equipped with a doubling measure supporting a p-Poincaré inequality, with 1<p><∞. For bounded domains in unweighted Rn, the uniqueness was shown for the p-Laplace operator Δp and all p by Kichenassamy and Véron (1986) [25], while for p = 2 it is an easy consequence of the linearity of the Laplace operator Δ. Beyond that, uniqueness is only known in some particular cases, such as in Ahlfors p-regular spaces, as shown by Bonk et al. (2022) [10]. When the singularity x0 has positive p capacity, the Green function is a particular multiple of the capacitary potential for capp({x0},Ω) and is therefore unique. Here we give a sufficient condition for uniqueness in metric spaces, and provide an example showing that the range of p for which it holds (while x0 has zero p-capacity) can be a nondegenerate interval. In the opposite direction, we give the first example showing that uniqueness can fail in metric spaces, even for p = 2.</p>peerReviewe